Explicit n-descent on elliptic curves. II. Geometry - Jacobs University
EXPLICIT n-DESCENT ON ELLIPTIC CURVES II. GEOMETRY J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
Abstract. This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in Pn−1 . The main tool we use is a 2 comparison between an easily obtained embedding into Pn −1 and 2 another map into Pn −1 that factors through the Segre embedding 2 Pn−1 ×Pn−1 → Pn −1 . The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.
1. Introduction This paper is the second in a series of papers discussing ‘Explicit ndescent on elliptic curves.’ Let E be an elliptic curve over a number field K, and let n ≥ 3. The elements of the n-Selmer group of E may be viewed as isomorphism classes of n-coverings C → E. For us, to do an ‘explicit n-descent’ means to represent each such isomorphism class by giving equations for C as a curve of degree n in Pn−1 . It is well known that computing the n-Selmer group as an abstract group gives partial information about both the Mordell-Weil group E(K) and the Tate-Shafarevich group X(K, E). There are likewise several motivations for computing the n-Selmer group in the explicit sense described above. Firstly, it is known that a point in E(K) of xcoordinate height h lifts to a point of height approximately h/(2n) on one of the Selmer n-coverings (see Theorem B.3.2 in [6]). Thus explicit n-descent enables us to find generators for the Mordell-Weil group more easily, and hence sometimes to show that the n-torsion of the TateShafarevich group is trivial. Secondly, if we have already computed the Mordell-Weil group, for instance using descent at some other n0 , then we can use explicit n-descent to exhibit concrete examples of nontrivial n-torsion elements of X(K, E). Our work is also likely to form a Date: 20th November 2006. 1
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
useful starting point for performing higher descents, and for computing the Cassels-Tate pairing. The first paper in this series [3] gives more background on the theory of descent. In particular we explain a number of different interpretations of the elements of the group H 1 (K, E[n]), which contains the Selmer group as a subgroup. One of these interpretations, and the one which is relevant to this paper, is as ‘Brauer-Severi diagrams’ [C → S]; such a diagram consists of a morphism that is a twist of the embedding E → Pn−1 associated to the complete linear system |n(O)|, where O is the origin of E. In particular C is a torsor under E and S is a Brauer-Severi variety of dimension n − 1. If C has points everywhere locally, then so does S, hence (by Global Class Field Theory) S is isomorphic to Pn−1 . Our goal in this paper is to explain how one can obtain equations for the image of C in S ∼ = Pn−1 in this case. 2
In brief, we compare two different embeddings of C into Pn −1 . The first comes from the fact that, even without assuming C has points everywhere locally, the cohomology map H 1 (K, E[n]) → H 1 (K, E[n2 ]) sends the Brauer-Severi diagram [C → S] to a diagram [C → S 0 ] 2 with S 0 ∼ = Pn −1 . The second map is more abstract: starting with the diagram C → S we form a dual map C → S ∨ and thus C → S × S ∨ . We then compose with the (generalised) Segre embedding to obtain 2 −1
C → S × S ∨ → Pn
.
2
Equations for the image of the first map to Pn −1 are given in Section 3. 2 By a suitable change of coordinates on Pn −1 followed by projection to a hyperplane, we obtain equations for the image of the second map. Finally, we pull back to C → S and, when we start with a Selmer group element, obtain equations for C → Pn−1 . For obvious reasons we refer to this as the Segre embedding method. We will not be concerned in this paper with the details of implementation; these will be discussed in the third paper of the series [4]. However, we would like to mention that the Segre embedding method, as well as two more methods discussed in [3], have been implemented for n = 3 and K = Q and are available as part of the MAGMA computer algebra system [7] (version 2.13 or later). All three methods rely for their practical implementation on a ‘Black Box’ that computes, for a given central simple K-algebra A of dimension n2 that is known to be isomorphic to the matrix algebra Matn (K), an explicit isomorphism A → Matn (K). Algorithms for this when K = Q will be described in [4].
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
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2. Background and overview Unless stated otherwise, K will denote a number field, with absolute Galois group GK , and E will be an elliptic curve over K with origin O. Let n be a positive integer. Recall the definition of the n-Selmer group of E: the short exact sequence of K-group schemes [n]
0 −→ E[n] −→ E −→ E −→ 0 gives rise to the following commutative diagram with exact rows. δ
E(K)
Q
v E(Kv )
δ
/ H 1 (K, E[n]) / H 1 (K, E) RRR RRR α RRR RRR R Q QR) 1 1 / / H (Kv , E[n]) H (Kv , E) v
v
Here, v runs through all places of K. The n-Selmer group Sel(n) (K, E) is then defined to be the kernel of α. The following notation and facts can be found in [3]. We assume that n ≥ 3. Then there is an embedding f : E → Pn−1 associated to the complete linear system |n(O)|. In fact, if n = 2 one would work with a double cover, in which case our algorithm still works with minor changes. Indeed it reduces to the classical number field method for 2-descent (see for example [2], Lecture 15). We can view an element of H 1 (K, E[n]) as a twist of the diagram f : E → Pn−1 , i.e., as a diagram of the form [C → S], where C is a torsor under E and S is a Brauer-Severi variety of dimension n − 1. We call such a diagram a Brauer-Severi diagram. In this interpretation, a diagram [C → S] corresponds to an element of the n-Selmer group if and only if C has points everywhere locally. The period-index obstruction map, defined in [9], is a quadratic map Obn : H 1 (K, E[n]) −→ Br(K)[n]. It sends the diagram [C → S] to the class of S in Br(K)[n]. We say an element ξ of H 1 (K, E[n]) has trivial obstruction if Obn (ξ) = 0, equivalently the corresponding diagram [C → S] has S ∼ = Pn−1 . Alternatively we can view an element ξ of H 1 (K, E[n]) as a pair (C, [D]) where [D] is a GK -invariant divisor class on C. The class [D] is represented by a rational divisor if and only if the element ξ has trivial obstruction. Our algorithm applies not only to elements of the Selmer group, but more generally to any element in H 1 (K, E[n]) with trivial obstruction.
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The natural map H 1 (K, E[n]) → H 1 (K, E[n2 ]) brings the pair (C, [D]) to the pair (C, [nD]). Moreover, composing the above with Obn2 gives the zero map: there is a commutative diagram H 1 (K, E[n])
H 1 (K, E[n2 ])
Obn
Obn2
/ Br(K)[n]
·n
/ Br(K)[n2 ]
In other words, the divisor class [nD] is always represented by a rational divisor of degree n2 , or equivalently, the above map takes any diagram 2 [C → S] to a diagram [C → Pn −1 ] with trivial obstruction. It is relatively easy to find equations for the image of this map. We do this in Section 3. Our basic question then is, starting with an element of H 1 (K, E[n]) with trivial obstruction, how do we reverse the map 2 −1
[C → Pn−1 ] 7−→ [C → Pn
]?
The diagram [C → S] naturally extends to give a map λC : C −→ S × S ∨ −→ P(A) where A is the obstruction algebra, the central simple K-algebra associated to the Brauer group element Obn (C → S) = [S] (see [10, p. 160]). In this paper we describe an algorithm for writing down both structure constants for A and equations for C as a curve of degree n2 2 in P(A) ∼ = Pn −1 . In fact we specify the equations in Section 3 and the structure constants in Section 4. In the case of trivial obstruction, we know that there exists an isomorphism of K-algebras A ∼ = Matn (K), which is called a trivialisation of A. Using this isomorphism, we may obtain equations for C in Pn−1 as a curve of degree n by projecting onto a column. The algorithm is split into parts, each of which (except the first) correspond to a piece of the ‘master diagram’ found in Theorem 8.1 below, which we reproduce here. C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨
Segre
gC
P(R)
∼ ϕρ
/ P(Aρ )
∼ τρ
/ P(Matn ) O proj / P(Matn )
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The first step is to realize H 1 (K, E[n]) as a subgroup of a concrete group. Let R be the affine algebra of the group scheme E[n]. Addition in E[n] translates into the comultiplication homomorphism ∆ : R → R ⊗K R. We use this to define a group homomorphism α⊗α ∂ : R× −→ (R ⊗K R)× , α 7−→ . ∆(α) Then there is a natural embedding of H 1 (K, E[n]) into (R⊗K R)× /∂R× . See Section 3 below and [3, Section 3]. We can then compute the Selmer group as a subgroup of (R ⊗K R)× /∂R× , see [4] for details. Therefore, in the following we can assume that our element of H 1 (K, E[n]) is represented by some ρ ∈ (R ⊗K R)× . In the second step, starting with ρ ∈ (R ⊗K R)× coming from an element of H 1 (K, E[n]), we construct an embedding gC : C → P(R), where P(R) is the projective space associated to the K-vector space R. We need to choose a K-basis of R in order to write down explicit equations for the image of gC . This step is explained in Section 3. In the third step, we define a new multiplication on R, which depends on ρ, that turns it into a central simple K-algebra Aρ . This is the obstruction algebra for the element of H 1 (K, E[n]) represented by ρ. In other words, we create an explicit isomorphism of K-vector spaces ϕρ : R → Aρ . This will be explained in Section 4. The fourth step makes use of the fact that S ∼ = Pn−1 , or equivalently that Aρ ∼ = Matn (K), when ρ represents an element of H 1 (K, E[n]) which has trivial obstruction. In Section 5, we give an explicit trivialisation of the algebra A1 , the central simple algebra coming from the trivial element of H 1 (K, E[n]), which serves as a normalisation. For the general case a trivialisation map τρ : Aρ → Matn (K) will come from our ‘Black Box’, to be described in more detail in the third paper in this series [4]. In the fifth step, we project the image of τρ ◦ϕρ ◦gC into the hyperplane of trace zero matrices and show that our total map factors through the Segre embedding: Segre
C −→ Pn−1 × (Pn−1 )∨ −→ P(Matn ). This step makes up Section 6 for the case ρ = 1, i.e. [C → Pn−1 ] ∼ = [E → Pn−1 ]. The general case is described in Sections 7 and 8. The final step of the algorithm is to make use of the Segre factorisation. It is here that an explicit trivialisation of the obstruction algebra is required. We pull back under the Segre embedding and project to the
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first factor, which gives us equations for C → Pn−1 . This is explained in Section 8. 2 −1
3. Finding Equations for C in Pn
We continue to take K a number field, but in fact our algorithm works over any field whose characteristic is prime to n (although below, we assume for simplicity that the characteristic is neither 2 nor 3). We denote by [n] the multiplication-by-n map on E. We fix a Weierstrass equation E : y 2 = x3 + ax + b × ¯ a rational function r(T ,T ) in K(E) ¯ and define for T1 , T2 ∈ E[n](K) , 1 2 if T1 = O or T2 = O 1 x(P ) − x(T1 ) if T1 + T2 = O and T1 6= O r(T1 ,T2 ) (P ) = y(P )+y(T +T ) 1 2 − λ(T1 , T2 ) otherwise, x(P )−x(T1 +T2 ) where λ(T1 , T2 ) denotes the slope of the line joining T1 and T2 , respectively of the tangent line at T1 = T2 if the points are equal. ¯ there exists a rational function GT ∈ K(E) ¯ For T ∈ E[n](K), with divisor X X div(GT ) = (S) − (S) = [n]∗ (T ) − [n]∗ (O). nS=T
nS=O
(See [11, Section III.8].) Proposition 3.1. We can scale the {GT } such that (i) The map T 7→ GT is GK -equivariant, 2 ¯ ¯ and P ∈ E(K)\E[n ¯ (ii) For each T1 , T2 ∈ E[n](K) ](K) we have r(T1 ,T2 ) (nP ) =
GT1 (P )GT2 (P ) . GT1 +T2 (P )
(iii) GO = 1, and for T 6= O the residue of GT at O with respect to the local parameter x/y is n1 . Proof. Define the scalings of the GT so that condition (iii) holds. That means that with respect to the local parameter t = x/y, when T 6= O, we can write GT (t) = n1 t−1 + . . . , where ‘. . . ’ signifies ‘higher order terms.’ This choice of scaling makes the map T 7→ GT visibly GK equivariant.
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When T1 = O or T2 = O, condition (ii) holds trivially. In any case, a calculation of divisors gives div(r(T1 ,T2 ) ) = (T1 ) + (T2 ) − (O) − (T1 + T2 ) for all choices of T1 and T2 . So the divisor of r(T1 ,T2 ) ◦ [n] is X X X X div(r(T1 ,T2 ) ◦ [n]) = (S) + (S) − (S) − nS=T1
nS=T2
nS=O
(S).
nS=T1 +T2
This is exactly the divisor of P 7→ GT1 (P )GT2 (P )/GT1 +T2 (P ). Therefore condition (ii) holds up to a scalar. In the following, we consider the case T1 + T2 6= O. We write x(t) = t−2 + . . . and y(t) = t−3 + . . . . Then locally at O, y(t) + y(T1 + T2 ) r(T1 ,T2 ) (t) = − λ(T1 , T2 ) x(t) − x(T1 + T2 ) t−3 + . . . = −2 − λ(T1 , T2 ) t + ... = t−1 + . . . Next, from [11, Prop. IV.2.3], we have [n](t) = nt + . . . , hence 1 r(T1 ,T2 ) ◦ [n](t) = (nt)−1 + · · · = t−1 + . . . n Comparing this with ( n1 t−1 + . . . )( n1 t−1 + . . . ) GT1 (t)GT2 (t) 1 = = t−1 + . . . 1 −1 GT1 +T2 (t) n t + ... n shows that the scalar is 1. The case T1 + T2 = O is similar.
2
In preparation for defining the embedding of C in Pn −1 , we recall some facts from [3]. Let R be the affine algebra of E[n], i.e., ¯ K). ¯ R = MapK (E[n](K), It is isomorphic to a product of (finite) field extensions of K, one for ¯ We also work with the algebra each GK -orbit in E[n](K). ¯ = R ⊗K K ¯ = Map(E[n](K), ¯ K). ¯ R The Weil pairing en : E[n] × E[n] → µn determines an injection ¯ ,→ R ¯ × = Map(E[n](K), ¯ K ¯ ×) w : E[n](K) via w(S)(T ) = en (S, T ). ¯ × → (R ¯ ⊗K¯ R) ¯ × via As in Section 2, we define ∂ : R α⊗α α(T1 )α(T2 ) (1) ∂α = , i.e., (∂α)(T1 , T2 ) = ; ∆(α) α(T1 + T2 )
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
then there is an exact sequence (cf. [3, Section 3]) (2)
w ∂ ¯ −→ ¯ × −→ ¯ ⊗K¯ R) ¯ ×. 0 −→ E[n](K) R (R
For V a vector space over K, we write P(V ) = Proj(K[V ]), where K[V ] = ⊕d≥0 Symd (V ∗ ) is the ring of polynomial functions on V . We define R = ResR/K (A1 ), or equivalently, R = Spec(K[R]). For any K-scheme X, we have R(X) = A1 (Spec(R) ×Spec(K) X). In particular, R(L) = R ⊗K L for any field extension L/K. We also define R× = ResR/K (Gm ) and S × = ResR⊗K R/K (Gm ). These schemes inherit a multiplication from Gm . The groups of K-rational points are R× (K) = R× and S × (K) = (R ⊗K R)× . We may identify R× with an open subscheme of R. With this notation, the exact sequence of GK -modules (2), becomes an exact sequence of K-group schemes: (3)
w
∂
0 −→ E[n] −→ R× −→ S × .
Part (i) of Proposition 3.1 allows us to package the functions GT to ¯ to the class form a scheme map gE : E → P(R) sending P ∈ E(K) of the map T 7→ GT (P ). Away from the subscheme E[n2 ], the map gE can be lifted to a map gE0 to R× . Then we have a commutative diagram: 0 gE
2 E \ E[n ]
_
E
gE
/ R× / P(R)
Next, we use the GK -equivariance of (T1 , T2 ) 7→ r(T1 ,T2 ) to package the functions r(T1 ,T2 ) to form a scheme map r : E \ E[n] → S × sending ¯ \ E[n](K) ¯ to the map r(P ) : (T1 , T2 ) 7→ r(T ,T ) (P ). P ∈ E(K) 1 2 Proposition 3.2. The following diagram commutes. E \ E[n2 ] 0 gE
[n]
/ E \ E[n] r
R×
∂
/ S×
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
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¯ \ E[n2 ](K) ¯ and observe Proof. We take a geometric point P ∈ E(K) that GT1 (P )GT2 (P ) . ∂(gE0 (P ))(T1 , T2 ) = GT1 +T2 (P ) By Proposition 3.1(ii), this equals r(T1 ,T2 ) (nP ). ¯ we denote by zT the coordinate function on R×Spec(K) For T ∈ E[n](K) Spec(K(T )) given by evaluating at T , so zT (α) = α(T ). Proposition 3.3. Given a Weierstrass equation for E, we can explicitly compute a set of n2 (n2 − 3)/2 linearly independent quadrics over K which define the image of 2 gE : E −→ P(R) ∼ = Pn −1 . ¯ = E[n](K), then the zT are coordinate functions on R, and If E[n](K) the defining quadrics can be split into two groups as follows. For all ¯ \ {O}, we have T1 , T2 ∈ E[n](K) 2 x(T1 ) − x(T2 ) zO + zT1 z−T1 − zT2 z−T2 , ¯ \ {O} such that and for all T11 , T12 , T21 , T22 ∈ E[n](K) T11 + T12 = T21 + T22 = T 6= O, we have λ(T21 , T22 ) − λ(T11 , T12 ) zO zT − zT11 zT12 + zT21 zT22 . Proof. We first note that the GT are linearly independent. This follows from the fact they are eigenfunctions for distinct characters with respect to the action of E[n] by translation. (We are using the definition of the Weil pairing in [11, Section III.8].) Since there are n2 functions GT , they form a basis for the Riemann-Roch space of the 2 divisor [n]∗ (O). Hence gE embeds E into P(R) ∼ = Pn −1 as an elliptic normal curve of degree n2 . ¯ \ E[n2 ](K), ¯ and let z = g 0 (P ) ∈ R× (K) ¯ be Now let P ∈ E(K) E projective coordinates for gE (P ). By Proposition 3.2, we then have r(nP ) = ∂z, or equivalently, r(nP )∆(z) = z ⊗ z. We wish to eliminate P from this equation. Since zO (gE0 (P )) = 1, we can make the equation homogeneous by multiplying the left hand side with z(O). This gives r(nP ) z(O)∆(z) = z ⊗ z. Writing everything out in terms of a K-basis of R, we obtain n4 quadrics, some of whose coefficients involve rational functions of nP . We can eliminate these rational functions by linear algebra over K, to obtain a set of quadrics in K[R] — this is what we do in practice. In order to
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determine the dimension of the space spanned by them and the geom¯ In fact, it is etry of the object defined by them, we can work over K. sufficient to work over L = K(E[n]). Over L, the coordinate functions zT are defined. In terms of these, our system of equations is r(T1 ,T2 ) (nP ) zO zT1 +T2 = zT1 zT2 , ¯ × E[n](K). ¯ parametrised by (T1 , T2 ) ∈ E[n](K) If T1 = O or T2 = O, this reduces to a tautology. If T1 = T 6= O and T2 = −T , then we get 2 x(nP ) − x(T ) zO = zT z−T . We can eliminate x(nP ) by taking differences. Taking into account the symmetry T ↔ −T , this gives us d1 independent quadrics, where ( E[n](K) ¯ \ {O} (n2 − 3)/2 if n is odd −1= d1 = # {±1} n2 /2 if n is even. If T1 + T2 = T 6= O and T1 , T2 6= O, we obtain y(nP ) + y(T ) − λ(T1 , T2 ) zO zT = zT1 zT2 . x(nP ) − x(T ) Fixing T , we can again eliminate the dependence on P by taking differences. Taking into account the symmetry (T1 , T2 ) ↔ (T2 , T1 ), this provides us with d2 independent quadrics, where {(T , T ) : T , T , T + T 6= O} 1 2 1 2 1 2 ¯ \ {O} d2 = # − # E[n](K) (T1 , T2 ) ∼ (T2 , T1 ) ( (n2 − 1)(n2 − 3)/2 if n is odd = n2 (n2 − 4)/2 if n is even. Together, we obtain d1 + d2 = n2 (n2 − 3)/2 independent quadrics in either case. We have found an n2 (n2 − 3)/2-dimensional space of quadrics vanishing on the image of gE . In general (see for example the Corollary to Theorem 8 in [8]) the homogeneous ideal of an elliptic normal curve of degree m ≥ 4 is generated by a vector space of quadrics of dimension m(m − 3)/2. Therefore, our quadrics define the image of E in P(R) under gE .
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
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By Section 3 of [3], we can identify H 1 (K, E[n]) with a subgroup of (R ⊗K R)× /∂R× . Specifically, we define an injective map H 1 (K, E[n]) −→ (R ⊗K R)× /∂R× ¯× by sending ξ ∈ H 1 (K, E[n]) to ρ ∂R× where ρ = ∂γ for some γ ∈ R such that w(ξσ ) = σ(γ)/γ for all σ ∈ GK . Definition 3.4. We let H denote the subgroup of (R ⊗K R)× that maps to the image of H 1 (K, E[n]) in (R ⊗K R)× /∂R× . Starting with a representative ρ ∈ H, we fix a choice of γ as above. Then γ determines a cocycle class ξ ∈ H 1 (K, E[n]). We let π : C → E be the twist of the trivial n-covering [n] : E → E by ξ. In other words, there is a genus 1 curve C defined over K and an isomorphism ¯ with σ(φ) ◦ φ−1 = τξσ (translation by φ : C → E defined over K ¯ for all σ ∈ GK . The covering map is then π = [n] ◦ φ. ξσ ∈ E[n](K)) It is easy to check that π is defined over K. Proposition 3.5. Given ρ and C as above, there are rational functions ¯ ¯ such that GT,C ∈ K(C), indexed by T ∈ E[n](K), (i) The divisor of GT,C is X X div(GT,C ) = (S) − (S) = π ∗ (T ) − π ∗ (O). π(S)=T
π(S)=O
(ii) The map T 7→ GT,C is GK -equivariant. (iii) The functions GT,C are scaled so that r(T1 ,T2 ) (π(P )) = ρ(T1 , T2 )
GT1 ,C (P )GT2 ,C (P ) . GT1 +T2 ,C (P )
(iv) We can package these GT,C to form morphisms of schemes gC : C → P(R)
gC0 : C \ π ∗ E[n] → R× .
and
¯ (v) The following diagram, with vertical maps defined over K, commutes: gC / P(R) C φ
E
gE
·γ
/ P(R)
¯ × as above and define Proof. We take γ ∈ R (4)
GT,C (P ) = γ(T )−1 GT (φ(P ))
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
¯ Since π = [n] ◦ φ, statement (i) is immediate from the for P ∈ C(K). corresponding statement for the GT . Next we check Galois equivariance. Since σ(φ) ◦ φ−1 = τξσ we have σ GT (φP ) = GσT φ(σP ) + ξσ = en (ξσ , σT )GσT φ(σP ) where the second equality is the definition of the Weil pairing in [11, Section III.8]. On the other hand, since σ(γ)/γ = w(ξσ ), we have σ(γ(T )) = w(ξσ )(σT )γ(σT ) = en (ξσ , σT )γ(σT ). We deduce that σ(GT,C )(σP ) = σ(GT,C (P )) = γ(σT )−1 GσT (φ(σP )) = GσT,C (σP ). This proves (ii). For (iii) we compute r(T1 ,T2 ) (π(P )) = r(T1 ,T2 ) (n · φ(P )) =
GT1 (φ(P ))GT2 (φ(P )) GT ,C (P )GT2 ,C (P ) = ρ(T1 , T2 ) 1 . GT1 +T2 (φ(P )) GT1 +T2 ,C (P )
Here the second equality comes from Proposition 3.1. The third equality follows from (4) and ∂γ = ρ. Statement (iv) is a formal consequence of (ii), and (v) then follows from (4). The proofs of the following propositions are very similar to those of Propositions 3.2 and 3.3, and so will be omitted. Proposition 3.6. The following diagram commutes.
0 gC
R×
/ E \ E[n]
π
C \ π ∗ E[n] ∂
/ S×
·ρ
r
/ S×
Proposition 3.7. Given a Weierstrass equation for E and an element ρ ∈ H, with corresponding n-covering π : C → E, we can explicitly compute a set of n2 (n2 − 3)/2 linearly independent quadrics over K which define the image of 2 gC : C −→ P(R) ∼ = Pn −1 .
¯ = E[n](K), then the zT are coordinate functions on R, and If E[n](K) the defining quadrics can be split into two groups as follows. For all ¯ \ {O}, we have T1 , T2 ∈ E[n](K) 2 x(T1 ) − x(T2 ) zO + ρ(T1 , −T1 )zT1 z−T1 − ρ(T2 , −T2 )zT2 z−T2 ,
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
13
¯ \ {O} such that and for all T11 , T12 , T21 , T22 ∈ E[n](K) T11 + T12 = T21 + T22 = T 6= O, we have λ(T21 , T22 ) − λ(T11 , T12 ) zO zT − ρ(T11 , T12 )zT11 zT12 + ρ(T21 , T22 )zT21 zT22 . 4. A Multiplication Table for the Obstruction Algebra ¯ there is a rational As noted in [11, Section III.8], for each T ∈ E[n](K) ¯ function FT ∈ K(E) with div(FT ) = n(T ) − n(O). In the last section we defined rational functions GT . We now scale the FT so that FT ◦ [n] = GnT . It is equivalent to demand that the leading coefficient of each FT , when expanded as a Laurent series in the local parameter x/y at O, should be 1. (See ‘Step 1’ in Section 5.3 of [3].) It turns out that the FT are rather easy to compute; this will be explained in [4]. ¯ ⊗K¯ R) ¯ × by Following ‘Step 2’ (loc. cit.), we now define ε ∈ (R ε(T1 , T2 ) =
FT1 +T2 (P ) . FT1 (P )FT2 (P − T1 )
¯ \ {O, T1 , T1 + T2 }. By the discussion in Section 3 for any P ∈ E(K) of [3], this does not depend on P and satisfies ε(T1 , T2 )ε(T2 , T1 )−1 = en (T1 , T2 ). Since the map defining ε is Galois-equivariant, we obtain an element ε ∈ (R ⊗K R)× . The subgroup H ⊂ (R ⊗K R)× was defined in Definition 3.4. Proposition 4.1. There is a map H −→ {central simple K-algebras of dimension n2 } sending ρ to an algebra Aρ such that Obn (ξ) = [Aρ ] ∈ Br(K)[n], 1
where ξ ∈ H (K, E[n]) is the element represented by ρ ∈ H. In particular, if ρ0 = ρ ∂z for some z ∈ R× , then [Aρ ] = [Aρ0 ]; in fact, the algebras Aρ and Aρ0 are isomorphic. Proof. In ‘Step 3’ (loc. cit.) we defined Aρ = (R, +, ∗ερ ), where ∗ερ is a new multiplication on R. To define it we view R ⊗K R as an R-algebra via the comultiplication ∆ : R → R ⊗K R. Recall that this is defined by ∆(α)(T1 , T2 ) = α(T1 + T2 ).
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
The corresponding trace map Tr : R ⊗K R −→ R P is given by (Tr z)(T ) = T1 +T2 =T z(T1 , T2 ). Then we define x ∗ερ y = Tr(ερ · x ⊗ y) for all x, y ∈ R. The stated properties of Aρ were established in Section 4 of [3]. If ρ0 = ρ ∂z, then the isomorphism Aρ0 → Aρ is given by α 7→ zα where the multiplication takes place in R. Definition 4.2. Let ϕρ : R → Aρ be the isomorphism of underlying K-vector spaces, inherent in the proof of Proposition 4.1. The construction in the proof provides us with explicit structure constants of Aρ in terms of a K-basis of R. In [4], we will discuss how to compute the structure constants in practice. 5. Trivialisation of the obstruction algebra Recall that when ρ ∈ R has trivial obstruction there is a trivialisation isomorphism τρ : Aρ → Matn (K), where Aρ is the central simple algebra in Proposition 4.1. Our algorithm will need to make this trivialisation explicit. In general this will be carried out by the ‘Black Box’ to be discussed further in [4]; however, when ρ = 1 we can write down a standard trivialisation τ1 : A1 → Matn (K). In fact τ1 will depend on a choice of morphism fE : E → Pn−1 determined by the complete linear system |n(O)|. We make this choice now. Let fE∨ : E → (Pn−1 )∨ be the dual map of fE , i.e., the map that ¯ to the osculating hyperplane at fE (P ). The elements takes P ∈ E(K) n−1 of P will be written as column vectors, and the elements of (Pn−1 )∨ ¯ there is a matrix MT ∈ GLn (K) ¯ as row vectors. For each T ∈ E[n](K) n−1 such that translation by T on E extends to the automorphism of P defined by MT . Proposition 5.1. We may fix the scalings of the MT ’s so that (5)
fE∨ (O) · MT−1 · fE (P ) FT (P ) = fE∨ (O) · fE (P )
where the FT ’s are the rational functions on E defined in Section 4. In particular, MO = I. Proof. The right hand side is well-defined (the undetermined scalings of fE∨ (O) and of fE (P ) cancel out) and has divisor n(T ) − n(O).
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
15
¯ be the characteristic function of T , i.e., the map that takes Let δT ∈ R ¯ to 0. It is clear that T to 1 but sends all other elements of E[n](K) ¯ ¯ ¯ the set of δT for T ∈ E[n](K) are a basis for R as a K-vector space. Definition 5.2. Recall the isomorphism ϕ1 : R → A1 of underlying K-vector spaces. Let τ1 : A1 → Matn (K) be the linear map of K-vector spaces given by X τ1 ϕ1 (α) = α(T )MT . ¯ T ∈E[n](K)
(Since T 7→ MT is GK -equivariant, the map τ1 , though a priori defined ¯ as a map of K-vector spaces, is GK -equivariant.) Note that τ1 sends ¯ ¯ ϕ1 (δT ) ∈ A1 ⊗K K to MT ∈ Matn (K). Proposition 5.3. The map τ1 : A1 → Matn (K) is an isomorphism of K-algebras. Proof. (See also [3, Prop. 5.8.(ii)].) By [3, Lemma 4.8] the set of MT ¯ form a basis for Matn (K). ¯ So it is clear that τ1 is an for T ∈ E[n](K) isomorphism of K-vector spaces. We must show that it is also a ring homomorphism. We recall that A1 = (R, +, ∗ε ). The new multiplication ∗ε extends to ¯ given by a multiplication on R δT1 ∗ε δT2 = Tr(ε · δT1 ⊗ δT2 ) = ε(T1 , T2 )δT1 +T2 . Applying τ1 to both sides, it is apparent that what we have to show is that MT1 MT2 = ε(T1 , T2 )MT1 +T2 ¯ for all T1 , T2 ∈ E[n](K). In any case it is clear that MT1 MT2 = c MT1 +T2 ¯ ×. for some constant c ∈ K Substituting T = T1 + T2 in (5), we get (6)
fE∨ (O) · MT−1 MT−1 · fE (P ) 2 1 . FT1 +T2 (P ) = c ∨ fE (O) · fE (P )
We arbitrarily lift fE and fE∨ to rational maps E 99K An . The definition of MT gives (7)
MT−1 · fE (P ) = h(P )fE (P − T1 ) 1
¯ for some rational function h ∈ K(E). Premultiplying by fE∨ (O) we get (8)
h(P ) =
fE∨ (O) · MT−1 · fE (P ) 1 . ∨ fE (O) · fE (P − T1 )
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
Substituting (7) into (6) gives FT1 +T2 (P ) = c
fE∨ (O) · MT−1 · fE (P − T ) 2 h(P ). ∨ fE (O) · fE (P )
Then by (5) and (8) we obtain FT1 +T2 (P ) = c FT2 (P − T1 )FT1 (P ). Comparing with the definition of ε in Section 4, it follows that c = ε(T1 , T2 ) as required. 6. The Segre Factorisation for E Recall our convention that for V a vector space over K, we write P(V ) = Proj(K[V ]) where K[V ] = ⊕d≥0 Symd (V ∗ ) is the ring of polynomial functions on V . We abbreviate P(Matn (K)) as P(Matn ). The K-points of P(Matn ) may be identified with the set Matn (K)/K × . In the last section we fixed a morphism fE : E → Pn−1 determined by the complete linear system |n(O)|, and wrote fE∨ : E → (Pn−1 )∨ for the dual map. We now consider the composite map fE ×f ∨
Segre
λE : E −→E Pn−1 × (Pn−1 )∨ −→ P(Matn ). We recall that we represent elements of Pn−1 as column vectors, and elements of (Pn−1 )∨ as row vectors. The Segre map is then given by matrix multiplication. In particular, its image is the locus of rank 1 matrices. Since each point of E lies on its own osculating hyperplane, ¯ we have f ∨ (P ) · fE (P ) = 0. Then for P ∈ E(K) E Tr(λE (P )) = Tr(fE (P ) · fE∨ (P )) = Tr(fE∨ (P ) · fE (P )) = 0. That is, the image of λE is contained in the locus of trace zero matrices, which is a hyperplane in P(Matn ). There is a direct sum decomposition Matn (K) = hIn i⊕{Tr = 0}. Note that the trace zero subspace contains all the matrices MT for T 6= O. We write proj for the rational map P(Matn ) 99K P(Matn ) induced by the second projection. We defined maps gE : E → P(R), ϕ1 : R → A1 and τ1 : A1 → Matn (K) in Sections 3, 4 and 5, respectively. Theorem 6.1. The following diagram commutes. λE
E
/ P(Matn ) O proj
gE
P(R)
∼ ϕ1
/ P(A1 )
∼ τ1
/ P(Matn )
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
17
The maps gE and τ1 were defined using the GT ’s and the MT ’s, respectively. Note that the matrices MT depend on our choice of the embedding fE : E → Pn−1 . The proof of Theorem 6.1 is based on the following result. ¯ \ E[n](K) ¯ we have Theorem 6.2. For P ∈ E(K) X λE (P ) = GT (P )MT . T 6=O
Let us show how Theorem 6.1 follows from this. Since the commutativ¯ ity of the diagram is a geometric question, we areP free to work over K. From the definitions we have τ1 ◦ ϕ1 ◦ gE (P ) = T GT (P )MT . Then composing P with the projection map to the trace zero subspace we get P 7→ T 6=O GT (P )MT , which equals λE (P ) by Theorem 6.2. The proof of Theorem 6.2 is split into a series of lemmas. The following ¯ notation will be throughout. Let Q1 , . . . , Qn be n points in E(K) Pused n n n−1 → P and and let P = i=1 Qi . We define morphisms hE : E h∨E : E n → (Pn−1 )∨ as follows. For the first map we put hE (Q1 , . . . , Qn ) = fE (P ). The second map takes (Q1 , . . . , Qn ) to the hyperplane meeting E (or rather the image of fE ) in the divisor (P − nQ1 ) + . . . + (P − nQn ). Notice that since this divisor has sum O, it is indeed linearly equivalent to the hyperplane section. The first lemma can be seen as a multi-variable generalisation of formula (5) in Proposition 5.1. ¯ \ E[n](K) ¯ then Lemma 6.3. If Q1 , . . . , Qn ∈ E(K) GT (Q1 ) . . . GT (Qn ) =
h∨E (Q1 , . . . , Qn ) · MT−1 · hE (Q1 , . . . , Qn ) h∨E (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn )
¯ for all T ∈ E[n](K). Proof. We view each side as a rational function on E n . The strategy of the proof is first to compare divisors, and then to check scalings by specialising to the case Q1 = Q2 = . . . = Qn . Let pri : E n → E be projection to the ith factor. The left hand side has divisor n X n X X X ∗ pri (x) − pr∗i (x). i=1 nx=T
i=1 nx=O
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
From the definitions of hE and h∨E the right hand side has a zero whenever nQi = T for some i, and a pole whenever nQi = O for some i. Therefore the right hand side has divisor n X X
ax pr∗i (x)
−
i=1 nx=T
n X X
bx pr∗i (x)
i=1 nx=O
where the ax and bx are positive integers. ¯ then the right hand side If we replace Q1 by Q1 + S for S ∈ E[n](K), is multiplied by a non-zero scalar (the commutator of MS and MT ). It follows that the integers ax and bx do not depend on x. So the right hand side has divisor n X n X X X ∗ a pri (x) − b pr∗i (x) i=1 nx=T
i=1 nx=O
where a and b are positive integers. Since this divisor is principal, its pull-back by any morphism E → E n has degree 0. This enables us to show that a = b. Since E n is a projective variety, it follows that the right hand side is c GT (Q1 )a · · · GT (Qn )a ¯ ×. for some constant c ∈ K We now specialise by taking Q1 = Q2 = . . . = Qn (= Q say). Then P = nQ and fE∨ (O) · MT−1 · fE (P ) na c GT (Q) = . fE∨ (O) · fE (P ) The definition of FT in Section 4 gives FT (P ) = FT (nQ) = GT (Q)n . Finally we compare with (5) to get c = 1 and a = 1. ¯ \ E[n](K) ¯ then Lemma 6.4. If Q1 , . . . , Qn ∈ E(K) 1 X hE (Q1 , . . . , Qn ) · h∨E (Q1 , . . . , Qn ) GT (Q1 ) . . . GT (Qn )MT = ∨ n hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) ¯ T ∈E[n](K)
¯ we can write the right Proof. Since the P MT form a basis for Matn (K), −1 hand side as n T aT (Q1 , . . . , Qn )MT for some rational functions aT n on E . To compute the aT ’s we premultiply by MT−1 and take the trace. Since ( n if T = O, Tr(MT ) = 0 otherwise,
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
19
on the left hand side we get GT (Q1 ) . . . GT (Qn ). On the right hand side, using Tr(AB) = Tr(BA) gives ∨ −1 hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) aT (Q1 , . . . , Qn ) = Tr MT · ∨ hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) h∨ (Q , . . . , Q ) · M −1 · h (Q , . . . , Q ) 1 n E 1 n T = Tr E ∨ hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) = GT (Q1 ) . . . GT (Qn ), where we have used Lemma 6.3 in the last equality.
Lemma 6.5. There is a commutative diagram of morphisms E
n
hE ×h∨ E
Q
P(R)
Segre
/ Pn−1 × (Pn−1 )∨
/ P(Matn )
gE
/ P(A1 )
∼ ϕ1
∼ τ1
/ P(Matn )
Q Q where ( gE )(Q1 , . . . , Qn ) = ni=1 gE (Qi ). Proof. From the definitions we have Y X τ1 ◦ ϕ1 ◦ ( gE )(Q1 , . . . , Qn ) = GT (Q1 ) . . . GT (Qn )MT . T
So the commutativity is already clear from Lemma 6.4. Q It only remains to check that gE is a morphism. To do this we write it as a composite gn
Q
E E n −→ P(R)n 99K P(R)
where the second map is induced by multiplication in R. We check that the image of the first map is contained in the domain of definition of the second. ¯ ¯ then gE (Q) is the class of T 7→ GT (Q), whereas If Q ∈ E(K)\E[n]( K), ¯ then gE (Q) is the class of T 7→ resQ (GT ), where the if Q ∈ E[n](K), residue is taken with respect to a local parameter at Q. (The choice of local parameter does not matter.) ¯ and Qn gE (Qi ) is undefined as an Now suppose Q1 , . . . , Qn ∈ E(K) i=1 element of P(R). Then for each T 6= O there exists 1 ≤ i ≤ n such that GT (Qi ) = 0, and hence nQi = T . But there are n2 − 1 such choices Q of T and only n choices of i. So this is impossible. It follows that gE is a morphism as claimed.
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
To complete the proof of Theorem 6.2, and hence of Theorem 6.1, we put Q1 = Q and Q2 = . . . = Qn = O in Lemma 6.5. Notice that hE (Q, O, . . . , O) = fE (Q) and h∨E (Q, O, . . . , O) = fE∨ (Q). Also, P with the GT ’s scaled as in Proposition 3.1(iii), gE (O) is the class of T 6=O δT . ¯ \ E[n](K) ¯ we obtain So for Q ∈ E(K) λE (Q) = Segre ◦ (fE × fE∨ )(Q) = Segre ◦ (hE × h∨E )(Q, O, . . . , O) = τ1 ◦ ϕ1 (gE (O)n−1 gE (Q)) X = GT (Q)MT T 6=O
7. The Segre Factorisation for C Let A be a central simple algebra over K of dimension n2 . Let S and S ∨ be the Brauer-Severi varieties given by the minimal right and left ideals of A, respectively (see [10, p. 160]). There is a natural map Segre : S × S ∨ → P(A) given by intersecting ideals. We say an element a ∈ A has rank r if the map x 7→ ax is an endomorphism of A (as a K-vector space) of rank rn. Lemma 7.1. The Segre map S × S ∨ → P(A) is an embedding, with image the locus of rank 1 elements in P(A). Proof. If A ∼ = Matn (K) then S ∼ = Pn−1 and the Segre map reduces to that studied in Section 6. The description of the image is no more than the observation that a (non-zero) matrix has rank 1 if and only if it can be written as a column vector times a row vector. In general we ¯ ¯ ∼ ¯ use that there is an isomorphism of K-algebras A ⊗K K = Matn (K). The Segre map is an embedding since, on the rank 1 locus in P(A), an inverse is given by sending the class of a ∈ A to the pair of ideals (aA, Aa). From now on, we call the Segre map the (generalised) Segre embedding. We write 1A for the multiplicative identity of A, and Trd : A → K for the reduced trace. There is a decomposition of K-vector spaces A = h1A i ⊕ {Trd = 0}. As in Section 6 we write proj for projection onto the second factor. The subgroup H ⊂ (R ⊗K R)× was defined in Definition 3.4. Theorem 7.2. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Let S and S ∨ be the Brauer-Severi varieties given by the
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
21
minimal right and left ideals in P(Aρ ). Then there is a morphism fC : C → S with dual fC∨ : C → S ∨ such that (i) the following diagram commutes: C
∨ fC ×fC
/ S × S∨
gC
P(R)
∼ ϕρ
Segre
/ P(Aρ ) O proj / P(Aρ )
(ii) the Brauer-Severi diagram [C → S] and the class of ρ in H correspond to the same element of H 1 (K, E[n]). The theorem is proved by combining Theorem 6.1 with the next lemma. First we recall how the element ρ ∈ H and n-covering π : C → E are ¯ × with ρ = ∂γ and related. In Section 3 we fixed an element γ ∈ R defined a cocycle ξ ∈ H 1 (K, E[n]) via w(ξσ ) = σ(γ)/γ for all σ ∈ GK . Then we let π : C → E be the ξ-twist of the trivial n-covering. Thus ¯ with π = [n] ◦ φ there is an isomorphism φ : C → E defined over K −1 and σ(φ) ◦ φ = τξσ for all σ ∈ GK . ¯ ¯ → Lemma 7.3. There is an isomorphism of K-algebras β : Aρ ⊗K K ¯ making the following diagram commute. A1 ⊗K K (9)
C
gC
φ
E
gE
/ P(R)
∼ ϕρ
·γ
/ P(R)
∼ ϕ1
proj
/ P(Aρ ) _ _ _ _/ P(Aρ ) KK LLL KKτ1 ◦β LLτL1 ◦β KK β LLL KK K L& % proj τ1 / P(A1 ) / P(Matn ) _ _ _/ P(Matn )
Proof. The first square commutes by Proposition 3.5(v). We define ¯ → A1 ⊗K K ¯ to make the second square commute. It β : Aρ ⊗K K ¯ is an isomorphism of K-algebras by [3, Lemma 4.6]. We recall from Proposition 5.3 that τ1 is an isomorphism of K-algebras. Finally, since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τ1 ◦ β. Theorem 6.1 identifies the composite of the second row of (9) as λE , where we recall fE ×f ∨
Segre
λE : E −→E Pn−1 × (Pn−1 )∨ −→ P(Matn ). Since λE factors via the Segre embedding, its image belongs to the rank 1 locus of P(Matn ). Let λC = proj ◦ϕρ ◦ gC be the composite
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
¯ of the first row of (9). Recalling that τ1 ◦ β is an isomorphism of Kalgebras, it follows that λC has image belonging to the rank 1 locus of P(Aρ ). By Lemma 7.1 it therefore factors via the Segre embedding. In other words, there are morphisms fC : C → S and fC∨ : C → S ∨ making the diagram in the first part of Theorem 7.2 commute. It remains to show that [C → S] is a Brauer-Severi diagram, that fC∨ is dual to fC and that [C → S] and ρ correspond to the same element of H 1 (K, E[n]). ¯ ¯ → Matn (K) ¯ induces The isomorphism of K-algebras τ1 ◦ β : Aρ ⊗K K n−1 ∨ ∨ n−1 ∨ ¯ isomorphisms ψ : S → P and ψ : S → (P ) defined over K making the following diagram commute. ∨ fC ×fC
C
ψ×ψ ∨
φ
E
∨ fE ×fE
Segre
/ S × S∨
Segre
/ Pn−1 × (Pn−1 )∨
/ P(Aρ )
τ1 ◦β
/ P(Matn )
This shows that via (φ, ψ), the morphism fC : C → S is isomorphic ¯ to fE : E → Pn−1 . Hence [C → S] is a Brauer-Severi diagram. over K Since σ(φ) ◦ φ−1 is translation by ξσ this diagram is the ξ-twist of [E → Pn−1 ], and therefore the Brauer-Severi diagram corresponding to ρ. At the same time, we see that fC∨ is dual to fC (since fE∨ is dual to fE ). This completes the proof of Theorem 7.2. 8. Finding Equations for C in Pn−1 We continue to represent elements of H 1 (K, E[n]) by elements ρ ∈ H, where the subgroup H ⊂ (R ⊗K R)× was defined in Section 3. The obstruction algebra Aρ was introduced in Section 4. If ρ represents an element of H 1 (K, E[n]) with trivial obstruction, then we make use of an explicit trivialisation isomorphism of K-algebras τρ : Aρ → Matn (K). Theorem 8.1. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Suppose given an isomorphism of K-algebras τρ : Aρ → Matn (K). Then there is a morphism fC0 : C → Pn−1 with dual fC0 ∨ such that (i) the following diagram commutes: C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨
Segre
gC
P(R)
∼ ϕρ
/ P(Aρ )
∼ τρ
/ P(Matn ) O proj / P(Matn )
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
23
(ii) the Brauer-Severi diagram [C → Pn−1 ] and the class of ρ in H correspond to the same element of H 1 (K, E[n]). Proof. Let S and S ∨ be the Brauer-Severi varieties given by the minimal right and left ideals in P(Aρ ). The isomorphism τρ induces Kisomorphisms ψ 0 : S → Pn−1 and ψ 0 ∨ : S ∨ → (Pn−1 )∨ . We modify the maps fC and fC∨ of Theorem 7.2 to give maps fC0 and fC0 ∨ making the following diagram commute.
C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨ O
Segre
ψ 0 ×ψ 0 ∨
C
∨ fC ×fC
/ S × S∨
/ P(Matn ) O τρ
Segre
/ P(Aρ )
Since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τρ . The theorem now follows by combining the above diagram with that in Theorem 7.2. In Section 3 we explained how to write down equations for C as a curve 2 of degree n2 in P(R) ∼ = Pn −1 . Theorem 8.1 now tells us how to convert these to equations for C as a curve of degree n in Pn−1 . First we use τρ to get equations for C in P(Matn ). Then we project onto the trace zero matrices. Next we write x11 , x12 , . . . , xnn for our coordinate functions on P(Matn ) and substitute xij = xi yj , where the xi and yj are new indeterminates (2n in total). This corresponds to pulling the image of C in P(Matn ) back under the Segre map. To project to the first factor Pn−1 , we eliminate the yj . We are left with equations in the xi , and these define the image of fC0 : C → Pn−1 . We will describe in [4] how this computation can be reduced to linear algebra for any specific value of n. For aesthetic as well as practical reasons, it is desirable to find a change of coordinates on Pn−1 so that the equations for C have small coefficients. The necessary minimisation and reduction procedures will be described in a forthcoming paper [5] for the cases n = 3 and n = 4. It is also desirable to have explicit equations for the covering map π : C → E. In principle these could be obtained using the methods of Section 3. However in the cases n = 2, 3, 4 equations for π are already given by classical formulae, reproduced in [1].
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References [1] S.Y. An, S.Y. Kim, D.C. Marshall, S.H. Marshall, W.G. McCallum and A.R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), no. 2, 304–315. [2] J.W.S. Cassels, Lectures on elliptic curves, LMS Student Texts 24, Cambridge University Press, Cambridge, 1991. [3] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, I Algebra, submitted for publication. [4] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, III Algorithms, in preparation. [5] J.E. Cremona, T.A. Fisher and M. Stoll, Minimisation and reduction for 3and 4-coverings of elliptic curves, in preparation. [6] M. Hindry and J.H. Silverman, Diophantine geometry, Graduate Texts in Mathematics 201, Springer-Verlag, New York, 2000. [7] MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24, 235–265 (1997). The Magma home page is at http://magma.maths.usyd.edu.au/magma/ [8] D. Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), pp. 29–100, Edizioni Cremonese, Rome, 1970. [9] C. O’Neil, The period-index obstruction for elliptic curves, J. Number Theory 95 (2002), no. 2, 329–339. [10] J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, New York, 1979. [11] J.H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK E-mail address: [email protected] University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK E-mail address: [email protected] Barnard College, Columbia University, Department of Mathematics, 2990 Broadway MC 4418, New York, NY 10027-6902,USA E-mail address: [email protected] ´ de Caen, Campus II - Boulevard Mare ´chal Juin, BP 5186– Universite 14032, Caen, France E-mail address: [email protected] School of Engineering and Science, International University Bremen (Jacobs University Bremen as of spring 2007), P.O. Box 750561, 28725 Bremen, Germany E-mail address: [email protected]
Abstract. This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in Pn−1 . The main tool we use is a 2 comparison between an easily obtained embedding into Pn −1 and 2 another map into Pn −1 that factors through the Segre embedding 2 Pn−1 ×Pn−1 → Pn −1 . The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.
1. Introduction This paper is the second in a series of papers discussing ‘Explicit ndescent on elliptic curves.’ Let E be an elliptic curve over a number field K, and let n ≥ 3. The elements of the n-Selmer group of E may be viewed as isomorphism classes of n-coverings C → E. For us, to do an ‘explicit n-descent’ means to represent each such isomorphism class by giving equations for C as a curve of degree n in Pn−1 . It is well known that computing the n-Selmer group as an abstract group gives partial information about both the Mordell-Weil group E(K) and the Tate-Shafarevich group X(K, E). There are likewise several motivations for computing the n-Selmer group in the explicit sense described above. Firstly, it is known that a point in E(K) of xcoordinate height h lifts to a point of height approximately h/(2n) on one of the Selmer n-coverings (see Theorem B.3.2 in [6]). Thus explicit n-descent enables us to find generators for the Mordell-Weil group more easily, and hence sometimes to show that the n-torsion of the TateShafarevich group is trivial. Secondly, if we have already computed the Mordell-Weil group, for instance using descent at some other n0 , then we can use explicit n-descent to exhibit concrete examples of nontrivial n-torsion elements of X(K, E). Our work is also likely to form a Date: 20th November 2006. 1
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
useful starting point for performing higher descents, and for computing the Cassels-Tate pairing. The first paper in this series [3] gives more background on the theory of descent. In particular we explain a number of different interpretations of the elements of the group H 1 (K, E[n]), which contains the Selmer group as a subgroup. One of these interpretations, and the one which is relevant to this paper, is as ‘Brauer-Severi diagrams’ [C → S]; such a diagram consists of a morphism that is a twist of the embedding E → Pn−1 associated to the complete linear system |n(O)|, where O is the origin of E. In particular C is a torsor under E and S is a Brauer-Severi variety of dimension n − 1. If C has points everywhere locally, then so does S, hence (by Global Class Field Theory) S is isomorphic to Pn−1 . Our goal in this paper is to explain how one can obtain equations for the image of C in S ∼ = Pn−1 in this case. 2
In brief, we compare two different embeddings of C into Pn −1 . The first comes from the fact that, even without assuming C has points everywhere locally, the cohomology map H 1 (K, E[n]) → H 1 (K, E[n2 ]) sends the Brauer-Severi diagram [C → S] to a diagram [C → S 0 ] 2 with S 0 ∼ = Pn −1 . The second map is more abstract: starting with the diagram C → S we form a dual map C → S ∨ and thus C → S × S ∨ . We then compose with the (generalised) Segre embedding to obtain 2 −1
C → S × S ∨ → Pn
.
2
Equations for the image of the first map to Pn −1 are given in Section 3. 2 By a suitable change of coordinates on Pn −1 followed by projection to a hyperplane, we obtain equations for the image of the second map. Finally, we pull back to C → S and, when we start with a Selmer group element, obtain equations for C → Pn−1 . For obvious reasons we refer to this as the Segre embedding method. We will not be concerned in this paper with the details of implementation; these will be discussed in the third paper of the series [4]. However, we would like to mention that the Segre embedding method, as well as two more methods discussed in [3], have been implemented for n = 3 and K = Q and are available as part of the MAGMA computer algebra system [7] (version 2.13 or later). All three methods rely for their practical implementation on a ‘Black Box’ that computes, for a given central simple K-algebra A of dimension n2 that is known to be isomorphic to the matrix algebra Matn (K), an explicit isomorphism A → Matn (K). Algorithms for this when K = Q will be described in [4].
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
3
2. Background and overview Unless stated otherwise, K will denote a number field, with absolute Galois group GK , and E will be an elliptic curve over K with origin O. Let n be a positive integer. Recall the definition of the n-Selmer group of E: the short exact sequence of K-group schemes [n]
0 −→ E[n] −→ E −→ E −→ 0 gives rise to the following commutative diagram with exact rows. δ
E(K)
Q
v E(Kv )
δ
/ H 1 (K, E[n]) / H 1 (K, E) RRR RRR α RRR RRR R Q QR) 1 1 / / H (Kv , E[n]) H (Kv , E) v
v
Here, v runs through all places of K. The n-Selmer group Sel(n) (K, E) is then defined to be the kernel of α. The following notation and facts can be found in [3]. We assume that n ≥ 3. Then there is an embedding f : E → Pn−1 associated to the complete linear system |n(O)|. In fact, if n = 2 one would work with a double cover, in which case our algorithm still works with minor changes. Indeed it reduces to the classical number field method for 2-descent (see for example [2], Lecture 15). We can view an element of H 1 (K, E[n]) as a twist of the diagram f : E → Pn−1 , i.e., as a diagram of the form [C → S], where C is a torsor under E and S is a Brauer-Severi variety of dimension n − 1. We call such a diagram a Brauer-Severi diagram. In this interpretation, a diagram [C → S] corresponds to an element of the n-Selmer group if and only if C has points everywhere locally. The period-index obstruction map, defined in [9], is a quadratic map Obn : H 1 (K, E[n]) −→ Br(K)[n]. It sends the diagram [C → S] to the class of S in Br(K)[n]. We say an element ξ of H 1 (K, E[n]) has trivial obstruction if Obn (ξ) = 0, equivalently the corresponding diagram [C → S] has S ∼ = Pn−1 . Alternatively we can view an element ξ of H 1 (K, E[n]) as a pair (C, [D]) where [D] is a GK -invariant divisor class on C. The class [D] is represented by a rational divisor if and only if the element ξ has trivial obstruction. Our algorithm applies not only to elements of the Selmer group, but more generally to any element in H 1 (K, E[n]) with trivial obstruction.
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
The natural map H 1 (K, E[n]) → H 1 (K, E[n2 ]) brings the pair (C, [D]) to the pair (C, [nD]). Moreover, composing the above with Obn2 gives the zero map: there is a commutative diagram H 1 (K, E[n])
H 1 (K, E[n2 ])
Obn
Obn2
/ Br(K)[n]
·n
/ Br(K)[n2 ]
In other words, the divisor class [nD] is always represented by a rational divisor of degree n2 , or equivalently, the above map takes any diagram 2 [C → S] to a diagram [C → Pn −1 ] with trivial obstruction. It is relatively easy to find equations for the image of this map. We do this in Section 3. Our basic question then is, starting with an element of H 1 (K, E[n]) with trivial obstruction, how do we reverse the map 2 −1
[C → Pn−1 ] 7−→ [C → Pn
]?
The diagram [C → S] naturally extends to give a map λC : C −→ S × S ∨ −→ P(A) where A is the obstruction algebra, the central simple K-algebra associated to the Brauer group element Obn (C → S) = [S] (see [10, p. 160]). In this paper we describe an algorithm for writing down both structure constants for A and equations for C as a curve of degree n2 2 in P(A) ∼ = Pn −1 . In fact we specify the equations in Section 3 and the structure constants in Section 4. In the case of trivial obstruction, we know that there exists an isomorphism of K-algebras A ∼ = Matn (K), which is called a trivialisation of A. Using this isomorphism, we may obtain equations for C in Pn−1 as a curve of degree n by projecting onto a column. The algorithm is split into parts, each of which (except the first) correspond to a piece of the ‘master diagram’ found in Theorem 8.1 below, which we reproduce here. C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨
Segre
gC
P(R)
∼ ϕρ
/ P(Aρ )
∼ τρ
/ P(Matn ) O proj / P(Matn )
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
5
The first step is to realize H 1 (K, E[n]) as a subgroup of a concrete group. Let R be the affine algebra of the group scheme E[n]. Addition in E[n] translates into the comultiplication homomorphism ∆ : R → R ⊗K R. We use this to define a group homomorphism α⊗α ∂ : R× −→ (R ⊗K R)× , α 7−→ . ∆(α) Then there is a natural embedding of H 1 (K, E[n]) into (R⊗K R)× /∂R× . See Section 3 below and [3, Section 3]. We can then compute the Selmer group as a subgroup of (R ⊗K R)× /∂R× , see [4] for details. Therefore, in the following we can assume that our element of H 1 (K, E[n]) is represented by some ρ ∈ (R ⊗K R)× . In the second step, starting with ρ ∈ (R ⊗K R)× coming from an element of H 1 (K, E[n]), we construct an embedding gC : C → P(R), where P(R) is the projective space associated to the K-vector space R. We need to choose a K-basis of R in order to write down explicit equations for the image of gC . This step is explained in Section 3. In the third step, we define a new multiplication on R, which depends on ρ, that turns it into a central simple K-algebra Aρ . This is the obstruction algebra for the element of H 1 (K, E[n]) represented by ρ. In other words, we create an explicit isomorphism of K-vector spaces ϕρ : R → Aρ . This will be explained in Section 4. The fourth step makes use of the fact that S ∼ = Pn−1 , or equivalently that Aρ ∼ = Matn (K), when ρ represents an element of H 1 (K, E[n]) which has trivial obstruction. In Section 5, we give an explicit trivialisation of the algebra A1 , the central simple algebra coming from the trivial element of H 1 (K, E[n]), which serves as a normalisation. For the general case a trivialisation map τρ : Aρ → Matn (K) will come from our ‘Black Box’, to be described in more detail in the third paper in this series [4]. In the fifth step, we project the image of τρ ◦ϕρ ◦gC into the hyperplane of trace zero matrices and show that our total map factors through the Segre embedding: Segre
C −→ Pn−1 × (Pn−1 )∨ −→ P(Matn ). This step makes up Section 6 for the case ρ = 1, i.e. [C → Pn−1 ] ∼ = [E → Pn−1 ]. The general case is described in Sections 7 and 8. The final step of the algorithm is to make use of the Segre factorisation. It is here that an explicit trivialisation of the obstruction algebra is required. We pull back under the Segre embedding and project to the
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
first factor, which gives us equations for C → Pn−1 . This is explained in Section 8. 2 −1
3. Finding Equations for C in Pn
We continue to take K a number field, but in fact our algorithm works over any field whose characteristic is prime to n (although below, we assume for simplicity that the characteristic is neither 2 nor 3). We denote by [n] the multiplication-by-n map on E. We fix a Weierstrass equation E : y 2 = x3 + ax + b × ¯ a rational function r(T ,T ) in K(E) ¯ and define for T1 , T2 ∈ E[n](K) , 1 2 if T1 = O or T2 = O 1 x(P ) − x(T1 ) if T1 + T2 = O and T1 6= O r(T1 ,T2 ) (P ) = y(P )+y(T +T ) 1 2 − λ(T1 , T2 ) otherwise, x(P )−x(T1 +T2 ) where λ(T1 , T2 ) denotes the slope of the line joining T1 and T2 , respectively of the tangent line at T1 = T2 if the points are equal. ¯ there exists a rational function GT ∈ K(E) ¯ For T ∈ E[n](K), with divisor X X div(GT ) = (S) − (S) = [n]∗ (T ) − [n]∗ (O). nS=T
nS=O
(See [11, Section III.8].) Proposition 3.1. We can scale the {GT } such that (i) The map T 7→ GT is GK -equivariant, 2 ¯ ¯ and P ∈ E(K)\E[n ¯ (ii) For each T1 , T2 ∈ E[n](K) ](K) we have r(T1 ,T2 ) (nP ) =
GT1 (P )GT2 (P ) . GT1 +T2 (P )
(iii) GO = 1, and for T 6= O the residue of GT at O with respect to the local parameter x/y is n1 . Proof. Define the scalings of the GT so that condition (iii) holds. That means that with respect to the local parameter t = x/y, when T 6= O, we can write GT (t) = n1 t−1 + . . . , where ‘. . . ’ signifies ‘higher order terms.’ This choice of scaling makes the map T 7→ GT visibly GK equivariant.
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
7
When T1 = O or T2 = O, condition (ii) holds trivially. In any case, a calculation of divisors gives div(r(T1 ,T2 ) ) = (T1 ) + (T2 ) − (O) − (T1 + T2 ) for all choices of T1 and T2 . So the divisor of r(T1 ,T2 ) ◦ [n] is X X X X div(r(T1 ,T2 ) ◦ [n]) = (S) + (S) − (S) − nS=T1
nS=T2
nS=O
(S).
nS=T1 +T2
This is exactly the divisor of P 7→ GT1 (P )GT2 (P )/GT1 +T2 (P ). Therefore condition (ii) holds up to a scalar. In the following, we consider the case T1 + T2 6= O. We write x(t) = t−2 + . . . and y(t) = t−3 + . . . . Then locally at O, y(t) + y(T1 + T2 ) r(T1 ,T2 ) (t) = − λ(T1 , T2 ) x(t) − x(T1 + T2 ) t−3 + . . . = −2 − λ(T1 , T2 ) t + ... = t−1 + . . . Next, from [11, Prop. IV.2.3], we have [n](t) = nt + . . . , hence 1 r(T1 ,T2 ) ◦ [n](t) = (nt)−1 + · · · = t−1 + . . . n Comparing this with ( n1 t−1 + . . . )( n1 t−1 + . . . ) GT1 (t)GT2 (t) 1 = = t−1 + . . . 1 −1 GT1 +T2 (t) n t + ... n shows that the scalar is 1. The case T1 + T2 = O is similar.
2
In preparation for defining the embedding of C in Pn −1 , we recall some facts from [3]. Let R be the affine algebra of E[n], i.e., ¯ K). ¯ R = MapK (E[n](K), It is isomorphic to a product of (finite) field extensions of K, one for ¯ We also work with the algebra each GK -orbit in E[n](K). ¯ = R ⊗K K ¯ = Map(E[n](K), ¯ K). ¯ R The Weil pairing en : E[n] × E[n] → µn determines an injection ¯ ,→ R ¯ × = Map(E[n](K), ¯ K ¯ ×) w : E[n](K) via w(S)(T ) = en (S, T ). ¯ × → (R ¯ ⊗K¯ R) ¯ × via As in Section 2, we define ∂ : R α⊗α α(T1 )α(T2 ) (1) ∂α = , i.e., (∂α)(T1 , T2 ) = ; ∆(α) α(T1 + T2 )
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
then there is an exact sequence (cf. [3, Section 3]) (2)
w ∂ ¯ −→ ¯ × −→ ¯ ⊗K¯ R) ¯ ×. 0 −→ E[n](K) R (R
For V a vector space over K, we write P(V ) = Proj(K[V ]), where K[V ] = ⊕d≥0 Symd (V ∗ ) is the ring of polynomial functions on V . We define R = ResR/K (A1 ), or equivalently, R = Spec(K[R]). For any K-scheme X, we have R(X) = A1 (Spec(R) ×Spec(K) X). In particular, R(L) = R ⊗K L for any field extension L/K. We also define R× = ResR/K (Gm ) and S × = ResR⊗K R/K (Gm ). These schemes inherit a multiplication from Gm . The groups of K-rational points are R× (K) = R× and S × (K) = (R ⊗K R)× . We may identify R× with an open subscheme of R. With this notation, the exact sequence of GK -modules (2), becomes an exact sequence of K-group schemes: (3)
w
∂
0 −→ E[n] −→ R× −→ S × .
Part (i) of Proposition 3.1 allows us to package the functions GT to ¯ to the class form a scheme map gE : E → P(R) sending P ∈ E(K) of the map T 7→ GT (P ). Away from the subscheme E[n2 ], the map gE can be lifted to a map gE0 to R× . Then we have a commutative diagram: 0 gE
2 E \ E[n ]
_
E
gE
/ R× / P(R)
Next, we use the GK -equivariance of (T1 , T2 ) 7→ r(T1 ,T2 ) to package the functions r(T1 ,T2 ) to form a scheme map r : E \ E[n] → S × sending ¯ \ E[n](K) ¯ to the map r(P ) : (T1 , T2 ) 7→ r(T ,T ) (P ). P ∈ E(K) 1 2 Proposition 3.2. The following diagram commutes. E \ E[n2 ] 0 gE
[n]
/ E \ E[n] r
R×
∂
/ S×
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
9
¯ \ E[n2 ](K) ¯ and observe Proof. We take a geometric point P ∈ E(K) that GT1 (P )GT2 (P ) . ∂(gE0 (P ))(T1 , T2 ) = GT1 +T2 (P ) By Proposition 3.1(ii), this equals r(T1 ,T2 ) (nP ). ¯ we denote by zT the coordinate function on R×Spec(K) For T ∈ E[n](K) Spec(K(T )) given by evaluating at T , so zT (α) = α(T ). Proposition 3.3. Given a Weierstrass equation for E, we can explicitly compute a set of n2 (n2 − 3)/2 linearly independent quadrics over K which define the image of 2 gE : E −→ P(R) ∼ = Pn −1 . ¯ = E[n](K), then the zT are coordinate functions on R, and If E[n](K) the defining quadrics can be split into two groups as follows. For all ¯ \ {O}, we have T1 , T2 ∈ E[n](K) 2 x(T1 ) − x(T2 ) zO + zT1 z−T1 − zT2 z−T2 , ¯ \ {O} such that and for all T11 , T12 , T21 , T22 ∈ E[n](K) T11 + T12 = T21 + T22 = T 6= O, we have λ(T21 , T22 ) − λ(T11 , T12 ) zO zT − zT11 zT12 + zT21 zT22 . Proof. We first note that the GT are linearly independent. This follows from the fact they are eigenfunctions for distinct characters with respect to the action of E[n] by translation. (We are using the definition of the Weil pairing in [11, Section III.8].) Since there are n2 functions GT , they form a basis for the Riemann-Roch space of the 2 divisor [n]∗ (O). Hence gE embeds E into P(R) ∼ = Pn −1 as an elliptic normal curve of degree n2 . ¯ \ E[n2 ](K), ¯ and let z = g 0 (P ) ∈ R× (K) ¯ be Now let P ∈ E(K) E projective coordinates for gE (P ). By Proposition 3.2, we then have r(nP ) = ∂z, or equivalently, r(nP )∆(z) = z ⊗ z. We wish to eliminate P from this equation. Since zO (gE0 (P )) = 1, we can make the equation homogeneous by multiplying the left hand side with z(O). This gives r(nP ) z(O)∆(z) = z ⊗ z. Writing everything out in terms of a K-basis of R, we obtain n4 quadrics, some of whose coefficients involve rational functions of nP . We can eliminate these rational functions by linear algebra over K, to obtain a set of quadrics in K[R] — this is what we do in practice. In order to
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
determine the dimension of the space spanned by them and the geom¯ In fact, it is etry of the object defined by them, we can work over K. sufficient to work over L = K(E[n]). Over L, the coordinate functions zT are defined. In terms of these, our system of equations is r(T1 ,T2 ) (nP ) zO zT1 +T2 = zT1 zT2 , ¯ × E[n](K). ¯ parametrised by (T1 , T2 ) ∈ E[n](K) If T1 = O or T2 = O, this reduces to a tautology. If T1 = T 6= O and T2 = −T , then we get 2 x(nP ) − x(T ) zO = zT z−T . We can eliminate x(nP ) by taking differences. Taking into account the symmetry T ↔ −T , this gives us d1 independent quadrics, where ( E[n](K) ¯ \ {O} (n2 − 3)/2 if n is odd −1= d1 = # {±1} n2 /2 if n is even. If T1 + T2 = T 6= O and T1 , T2 6= O, we obtain y(nP ) + y(T ) − λ(T1 , T2 ) zO zT = zT1 zT2 . x(nP ) − x(T ) Fixing T , we can again eliminate the dependence on P by taking differences. Taking into account the symmetry (T1 , T2 ) ↔ (T2 , T1 ), this provides us with d2 independent quadrics, where {(T , T ) : T , T , T + T 6= O} 1 2 1 2 1 2 ¯ \ {O} d2 = # − # E[n](K) (T1 , T2 ) ∼ (T2 , T1 ) ( (n2 − 1)(n2 − 3)/2 if n is odd = n2 (n2 − 4)/2 if n is even. Together, we obtain d1 + d2 = n2 (n2 − 3)/2 independent quadrics in either case. We have found an n2 (n2 − 3)/2-dimensional space of quadrics vanishing on the image of gE . In general (see for example the Corollary to Theorem 8 in [8]) the homogeneous ideal of an elliptic normal curve of degree m ≥ 4 is generated by a vector space of quadrics of dimension m(m − 3)/2. Therefore, our quadrics define the image of E in P(R) under gE .
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
11
By Section 3 of [3], we can identify H 1 (K, E[n]) with a subgroup of (R ⊗K R)× /∂R× . Specifically, we define an injective map H 1 (K, E[n]) −→ (R ⊗K R)× /∂R× ¯× by sending ξ ∈ H 1 (K, E[n]) to ρ ∂R× where ρ = ∂γ for some γ ∈ R such that w(ξσ ) = σ(γ)/γ for all σ ∈ GK . Definition 3.4. We let H denote the subgroup of (R ⊗K R)× that maps to the image of H 1 (K, E[n]) in (R ⊗K R)× /∂R× . Starting with a representative ρ ∈ H, we fix a choice of γ as above. Then γ determines a cocycle class ξ ∈ H 1 (K, E[n]). We let π : C → E be the twist of the trivial n-covering [n] : E → E by ξ. In other words, there is a genus 1 curve C defined over K and an isomorphism ¯ with σ(φ) ◦ φ−1 = τξσ (translation by φ : C → E defined over K ¯ for all σ ∈ GK . The covering map is then π = [n] ◦ φ. ξσ ∈ E[n](K)) It is easy to check that π is defined over K. Proposition 3.5. Given ρ and C as above, there are rational functions ¯ ¯ such that GT,C ∈ K(C), indexed by T ∈ E[n](K), (i) The divisor of GT,C is X X div(GT,C ) = (S) − (S) = π ∗ (T ) − π ∗ (O). π(S)=T
π(S)=O
(ii) The map T 7→ GT,C is GK -equivariant. (iii) The functions GT,C are scaled so that r(T1 ,T2 ) (π(P )) = ρ(T1 , T2 )
GT1 ,C (P )GT2 ,C (P ) . GT1 +T2 ,C (P )
(iv) We can package these GT,C to form morphisms of schemes gC : C → P(R)
gC0 : C \ π ∗ E[n] → R× .
and
¯ (v) The following diagram, with vertical maps defined over K, commutes: gC / P(R) C φ
E
gE
·γ
/ P(R)
¯ × as above and define Proof. We take γ ∈ R (4)
GT,C (P ) = γ(T )−1 GT (φ(P ))
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
¯ Since π = [n] ◦ φ, statement (i) is immediate from the for P ∈ C(K). corresponding statement for the GT . Next we check Galois equivariance. Since σ(φ) ◦ φ−1 = τξσ we have σ GT (φP ) = GσT φ(σP ) + ξσ = en (ξσ , σT )GσT φ(σP ) where the second equality is the definition of the Weil pairing in [11, Section III.8]. On the other hand, since σ(γ)/γ = w(ξσ ), we have σ(γ(T )) = w(ξσ )(σT )γ(σT ) = en (ξσ , σT )γ(σT ). We deduce that σ(GT,C )(σP ) = σ(GT,C (P )) = γ(σT )−1 GσT (φ(σP )) = GσT,C (σP ). This proves (ii). For (iii) we compute r(T1 ,T2 ) (π(P )) = r(T1 ,T2 ) (n · φ(P )) =
GT1 (φ(P ))GT2 (φ(P )) GT ,C (P )GT2 ,C (P ) = ρ(T1 , T2 ) 1 . GT1 +T2 (φ(P )) GT1 +T2 ,C (P )
Here the second equality comes from Proposition 3.1. The third equality follows from (4) and ∂γ = ρ. Statement (iv) is a formal consequence of (ii), and (v) then follows from (4). The proofs of the following propositions are very similar to those of Propositions 3.2 and 3.3, and so will be omitted. Proposition 3.6. The following diagram commutes.
0 gC
R×
/ E \ E[n]
π
C \ π ∗ E[n] ∂
/ S×
·ρ
r
/ S×
Proposition 3.7. Given a Weierstrass equation for E and an element ρ ∈ H, with corresponding n-covering π : C → E, we can explicitly compute a set of n2 (n2 − 3)/2 linearly independent quadrics over K which define the image of 2 gC : C −→ P(R) ∼ = Pn −1 .
¯ = E[n](K), then the zT are coordinate functions on R, and If E[n](K) the defining quadrics can be split into two groups as follows. For all ¯ \ {O}, we have T1 , T2 ∈ E[n](K) 2 x(T1 ) − x(T2 ) zO + ρ(T1 , −T1 )zT1 z−T1 − ρ(T2 , −T2 )zT2 z−T2 ,
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
13
¯ \ {O} such that and for all T11 , T12 , T21 , T22 ∈ E[n](K) T11 + T12 = T21 + T22 = T 6= O, we have λ(T21 , T22 ) − λ(T11 , T12 ) zO zT − ρ(T11 , T12 )zT11 zT12 + ρ(T21 , T22 )zT21 zT22 . 4. A Multiplication Table for the Obstruction Algebra ¯ there is a rational As noted in [11, Section III.8], for each T ∈ E[n](K) ¯ function FT ∈ K(E) with div(FT ) = n(T ) − n(O). In the last section we defined rational functions GT . We now scale the FT so that FT ◦ [n] = GnT . It is equivalent to demand that the leading coefficient of each FT , when expanded as a Laurent series in the local parameter x/y at O, should be 1. (See ‘Step 1’ in Section 5.3 of [3].) It turns out that the FT are rather easy to compute; this will be explained in [4]. ¯ ⊗K¯ R) ¯ × by Following ‘Step 2’ (loc. cit.), we now define ε ∈ (R ε(T1 , T2 ) =
FT1 +T2 (P ) . FT1 (P )FT2 (P − T1 )
¯ \ {O, T1 , T1 + T2 }. By the discussion in Section 3 for any P ∈ E(K) of [3], this does not depend on P and satisfies ε(T1 , T2 )ε(T2 , T1 )−1 = en (T1 , T2 ). Since the map defining ε is Galois-equivariant, we obtain an element ε ∈ (R ⊗K R)× . The subgroup H ⊂ (R ⊗K R)× was defined in Definition 3.4. Proposition 4.1. There is a map H −→ {central simple K-algebras of dimension n2 } sending ρ to an algebra Aρ such that Obn (ξ) = [Aρ ] ∈ Br(K)[n], 1
where ξ ∈ H (K, E[n]) is the element represented by ρ ∈ H. In particular, if ρ0 = ρ ∂z for some z ∈ R× , then [Aρ ] = [Aρ0 ]; in fact, the algebras Aρ and Aρ0 are isomorphic. Proof. In ‘Step 3’ (loc. cit.) we defined Aρ = (R, +, ∗ερ ), where ∗ερ is a new multiplication on R. To define it we view R ⊗K R as an R-algebra via the comultiplication ∆ : R → R ⊗K R. Recall that this is defined by ∆(α)(T1 , T2 ) = α(T1 + T2 ).
14
J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
The corresponding trace map Tr : R ⊗K R −→ R P is given by (Tr z)(T ) = T1 +T2 =T z(T1 , T2 ). Then we define x ∗ερ y = Tr(ερ · x ⊗ y) for all x, y ∈ R. The stated properties of Aρ were established in Section 4 of [3]. If ρ0 = ρ ∂z, then the isomorphism Aρ0 → Aρ is given by α 7→ zα where the multiplication takes place in R. Definition 4.2. Let ϕρ : R → Aρ be the isomorphism of underlying K-vector spaces, inherent in the proof of Proposition 4.1. The construction in the proof provides us with explicit structure constants of Aρ in terms of a K-basis of R. In [4], we will discuss how to compute the structure constants in practice. 5. Trivialisation of the obstruction algebra Recall that when ρ ∈ R has trivial obstruction there is a trivialisation isomorphism τρ : Aρ → Matn (K), where Aρ is the central simple algebra in Proposition 4.1. Our algorithm will need to make this trivialisation explicit. In general this will be carried out by the ‘Black Box’ to be discussed further in [4]; however, when ρ = 1 we can write down a standard trivialisation τ1 : A1 → Matn (K). In fact τ1 will depend on a choice of morphism fE : E → Pn−1 determined by the complete linear system |n(O)|. We make this choice now. Let fE∨ : E → (Pn−1 )∨ be the dual map of fE , i.e., the map that ¯ to the osculating hyperplane at fE (P ). The elements takes P ∈ E(K) n−1 of P will be written as column vectors, and the elements of (Pn−1 )∨ ¯ there is a matrix MT ∈ GLn (K) ¯ as row vectors. For each T ∈ E[n](K) n−1 such that translation by T on E extends to the automorphism of P defined by MT . Proposition 5.1. We may fix the scalings of the MT ’s so that (5)
fE∨ (O) · MT−1 · fE (P ) FT (P ) = fE∨ (O) · fE (P )
where the FT ’s are the rational functions on E defined in Section 4. In particular, MO = I. Proof. The right hand side is well-defined (the undetermined scalings of fE∨ (O) and of fE (P ) cancel out) and has divisor n(T ) − n(O).
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
15
¯ be the characteristic function of T , i.e., the map that takes Let δT ∈ R ¯ to 0. It is clear that T to 1 but sends all other elements of E[n](K) ¯ ¯ ¯ the set of δT for T ∈ E[n](K) are a basis for R as a K-vector space. Definition 5.2. Recall the isomorphism ϕ1 : R → A1 of underlying K-vector spaces. Let τ1 : A1 → Matn (K) be the linear map of K-vector spaces given by X τ1 ϕ1 (α) = α(T )MT . ¯ T ∈E[n](K)
(Since T 7→ MT is GK -equivariant, the map τ1 , though a priori defined ¯ as a map of K-vector spaces, is GK -equivariant.) Note that τ1 sends ¯ ¯ ϕ1 (δT ) ∈ A1 ⊗K K to MT ∈ Matn (K). Proposition 5.3. The map τ1 : A1 → Matn (K) is an isomorphism of K-algebras. Proof. (See also [3, Prop. 5.8.(ii)].) By [3, Lemma 4.8] the set of MT ¯ form a basis for Matn (K). ¯ So it is clear that τ1 is an for T ∈ E[n](K) isomorphism of K-vector spaces. We must show that it is also a ring homomorphism. We recall that A1 = (R, +, ∗ε ). The new multiplication ∗ε extends to ¯ given by a multiplication on R δT1 ∗ε δT2 = Tr(ε · δT1 ⊗ δT2 ) = ε(T1 , T2 )δT1 +T2 . Applying τ1 to both sides, it is apparent that what we have to show is that MT1 MT2 = ε(T1 , T2 )MT1 +T2 ¯ for all T1 , T2 ∈ E[n](K). In any case it is clear that MT1 MT2 = c MT1 +T2 ¯ ×. for some constant c ∈ K Substituting T = T1 + T2 in (5), we get (6)
fE∨ (O) · MT−1 MT−1 · fE (P ) 2 1 . FT1 +T2 (P ) = c ∨ fE (O) · fE (P )
We arbitrarily lift fE and fE∨ to rational maps E 99K An . The definition of MT gives (7)
MT−1 · fE (P ) = h(P )fE (P − T1 ) 1
¯ for some rational function h ∈ K(E). Premultiplying by fE∨ (O) we get (8)
h(P ) =
fE∨ (O) · MT−1 · fE (P ) 1 . ∨ fE (O) · fE (P − T1 )
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
Substituting (7) into (6) gives FT1 +T2 (P ) = c
fE∨ (O) · MT−1 · fE (P − T ) 2 h(P ). ∨ fE (O) · fE (P )
Then by (5) and (8) we obtain FT1 +T2 (P ) = c FT2 (P − T1 )FT1 (P ). Comparing with the definition of ε in Section 4, it follows that c = ε(T1 , T2 ) as required. 6. The Segre Factorisation for E Recall our convention that for V a vector space over K, we write P(V ) = Proj(K[V ]) where K[V ] = ⊕d≥0 Symd (V ∗ ) is the ring of polynomial functions on V . We abbreviate P(Matn (K)) as P(Matn ). The K-points of P(Matn ) may be identified with the set Matn (K)/K × . In the last section we fixed a morphism fE : E → Pn−1 determined by the complete linear system |n(O)|, and wrote fE∨ : E → (Pn−1 )∨ for the dual map. We now consider the composite map fE ×f ∨
Segre
λE : E −→E Pn−1 × (Pn−1 )∨ −→ P(Matn ). We recall that we represent elements of Pn−1 as column vectors, and elements of (Pn−1 )∨ as row vectors. The Segre map is then given by matrix multiplication. In particular, its image is the locus of rank 1 matrices. Since each point of E lies on its own osculating hyperplane, ¯ we have f ∨ (P ) · fE (P ) = 0. Then for P ∈ E(K) E Tr(λE (P )) = Tr(fE (P ) · fE∨ (P )) = Tr(fE∨ (P ) · fE (P )) = 0. That is, the image of λE is contained in the locus of trace zero matrices, which is a hyperplane in P(Matn ). There is a direct sum decomposition Matn (K) = hIn i⊕{Tr = 0}. Note that the trace zero subspace contains all the matrices MT for T 6= O. We write proj for the rational map P(Matn ) 99K P(Matn ) induced by the second projection. We defined maps gE : E → P(R), ϕ1 : R → A1 and τ1 : A1 → Matn (K) in Sections 3, 4 and 5, respectively. Theorem 6.1. The following diagram commutes. λE
E
/ P(Matn ) O proj
gE
P(R)
∼ ϕ1
/ P(A1 )
∼ τ1
/ P(Matn )
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
17
The maps gE and τ1 were defined using the GT ’s and the MT ’s, respectively. Note that the matrices MT depend on our choice of the embedding fE : E → Pn−1 . The proof of Theorem 6.1 is based on the following result. ¯ \ E[n](K) ¯ we have Theorem 6.2. For P ∈ E(K) X λE (P ) = GT (P )MT . T 6=O
Let us show how Theorem 6.1 follows from this. Since the commutativ¯ ity of the diagram is a geometric question, we areP free to work over K. From the definitions we have τ1 ◦ ϕ1 ◦ gE (P ) = T GT (P )MT . Then composing P with the projection map to the trace zero subspace we get P 7→ T 6=O GT (P )MT , which equals λE (P ) by Theorem 6.2. The proof of Theorem 6.2 is split into a series of lemmas. The following ¯ notation will be throughout. Let Q1 , . . . , Qn be n points in E(K) Pused n n n−1 → P and and let P = i=1 Qi . We define morphisms hE : E h∨E : E n → (Pn−1 )∨ as follows. For the first map we put hE (Q1 , . . . , Qn ) = fE (P ). The second map takes (Q1 , . . . , Qn ) to the hyperplane meeting E (or rather the image of fE ) in the divisor (P − nQ1 ) + . . . + (P − nQn ). Notice that since this divisor has sum O, it is indeed linearly equivalent to the hyperplane section. The first lemma can be seen as a multi-variable generalisation of formula (5) in Proposition 5.1. ¯ \ E[n](K) ¯ then Lemma 6.3. If Q1 , . . . , Qn ∈ E(K) GT (Q1 ) . . . GT (Qn ) =
h∨E (Q1 , . . . , Qn ) · MT−1 · hE (Q1 , . . . , Qn ) h∨E (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn )
¯ for all T ∈ E[n](K). Proof. We view each side as a rational function on E n . The strategy of the proof is first to compare divisors, and then to check scalings by specialising to the case Q1 = Q2 = . . . = Qn . Let pri : E n → E be projection to the ith factor. The left hand side has divisor n X n X X X ∗ pri (x) − pr∗i (x). i=1 nx=T
i=1 nx=O
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
From the definitions of hE and h∨E the right hand side has a zero whenever nQi = T for some i, and a pole whenever nQi = O for some i. Therefore the right hand side has divisor n X X
ax pr∗i (x)
−
i=1 nx=T
n X X
bx pr∗i (x)
i=1 nx=O
where the ax and bx are positive integers. ¯ then the right hand side If we replace Q1 by Q1 + S for S ∈ E[n](K), is multiplied by a non-zero scalar (the commutator of MS and MT ). It follows that the integers ax and bx do not depend on x. So the right hand side has divisor n X n X X X ∗ a pri (x) − b pr∗i (x) i=1 nx=T
i=1 nx=O
where a and b are positive integers. Since this divisor is principal, its pull-back by any morphism E → E n has degree 0. This enables us to show that a = b. Since E n is a projective variety, it follows that the right hand side is c GT (Q1 )a · · · GT (Qn )a ¯ ×. for some constant c ∈ K We now specialise by taking Q1 = Q2 = . . . = Qn (= Q say). Then P = nQ and fE∨ (O) · MT−1 · fE (P ) na c GT (Q) = . fE∨ (O) · fE (P ) The definition of FT in Section 4 gives FT (P ) = FT (nQ) = GT (Q)n . Finally we compare with (5) to get c = 1 and a = 1. ¯ \ E[n](K) ¯ then Lemma 6.4. If Q1 , . . . , Qn ∈ E(K) 1 X hE (Q1 , . . . , Qn ) · h∨E (Q1 , . . . , Qn ) GT (Q1 ) . . . GT (Qn )MT = ∨ n hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) ¯ T ∈E[n](K)
¯ we can write the right Proof. Since the P MT form a basis for Matn (K), −1 hand side as n T aT (Q1 , . . . , Qn )MT for some rational functions aT n on E . To compute the aT ’s we premultiply by MT−1 and take the trace. Since ( n if T = O, Tr(MT ) = 0 otherwise,
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
19
on the left hand side we get GT (Q1 ) . . . GT (Qn ). On the right hand side, using Tr(AB) = Tr(BA) gives ∨ −1 hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) aT (Q1 , . . . , Qn ) = Tr MT · ∨ hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) h∨ (Q , . . . , Q ) · M −1 · h (Q , . . . , Q ) 1 n E 1 n T = Tr E ∨ hE (Q1 , . . . , Qn ) · hE (Q1 , . . . , Qn ) = GT (Q1 ) . . . GT (Qn ), where we have used Lemma 6.3 in the last equality.
Lemma 6.5. There is a commutative diagram of morphisms E
n
hE ×h∨ E
Q
P(R)
Segre
/ Pn−1 × (Pn−1 )∨
/ P(Matn )
gE
/ P(A1 )
∼ ϕ1
∼ τ1
/ P(Matn )
Q Q where ( gE )(Q1 , . . . , Qn ) = ni=1 gE (Qi ). Proof. From the definitions we have Y X τ1 ◦ ϕ1 ◦ ( gE )(Q1 , . . . , Qn ) = GT (Q1 ) . . . GT (Qn )MT . T
So the commutativity is already clear from Lemma 6.4. Q It only remains to check that gE is a morphism. To do this we write it as a composite gn
Q
E E n −→ P(R)n 99K P(R)
where the second map is induced by multiplication in R. We check that the image of the first map is contained in the domain of definition of the second. ¯ ¯ then gE (Q) is the class of T 7→ GT (Q), whereas If Q ∈ E(K)\E[n]( K), ¯ then gE (Q) is the class of T 7→ resQ (GT ), where the if Q ∈ E[n](K), residue is taken with respect to a local parameter at Q. (The choice of local parameter does not matter.) ¯ and Qn gE (Qi ) is undefined as an Now suppose Q1 , . . . , Qn ∈ E(K) i=1 element of P(R). Then for each T 6= O there exists 1 ≤ i ≤ n such that GT (Qi ) = 0, and hence nQi = T . But there are n2 − 1 such choices Q of T and only n choices of i. So this is impossible. It follows that gE is a morphism as claimed.
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
To complete the proof of Theorem 6.2, and hence of Theorem 6.1, we put Q1 = Q and Q2 = . . . = Qn = O in Lemma 6.5. Notice that hE (Q, O, . . . , O) = fE (Q) and h∨E (Q, O, . . . , O) = fE∨ (Q). Also, P with the GT ’s scaled as in Proposition 3.1(iii), gE (O) is the class of T 6=O δT . ¯ \ E[n](K) ¯ we obtain So for Q ∈ E(K) λE (Q) = Segre ◦ (fE × fE∨ )(Q) = Segre ◦ (hE × h∨E )(Q, O, . . . , O) = τ1 ◦ ϕ1 (gE (O)n−1 gE (Q)) X = GT (Q)MT T 6=O
7. The Segre Factorisation for C Let A be a central simple algebra over K of dimension n2 . Let S and S ∨ be the Brauer-Severi varieties given by the minimal right and left ideals of A, respectively (see [10, p. 160]). There is a natural map Segre : S × S ∨ → P(A) given by intersecting ideals. We say an element a ∈ A has rank r if the map x 7→ ax is an endomorphism of A (as a K-vector space) of rank rn. Lemma 7.1. The Segre map S × S ∨ → P(A) is an embedding, with image the locus of rank 1 elements in P(A). Proof. If A ∼ = Matn (K) then S ∼ = Pn−1 and the Segre map reduces to that studied in Section 6. The description of the image is no more than the observation that a (non-zero) matrix has rank 1 if and only if it can be written as a column vector times a row vector. In general we ¯ ¯ ∼ ¯ use that there is an isomorphism of K-algebras A ⊗K K = Matn (K). The Segre map is an embedding since, on the rank 1 locus in P(A), an inverse is given by sending the class of a ∈ A to the pair of ideals (aA, Aa). From now on, we call the Segre map the (generalised) Segre embedding. We write 1A for the multiplicative identity of A, and Trd : A → K for the reduced trace. There is a decomposition of K-vector spaces A = h1A i ⊕ {Trd = 0}. As in Section 6 we write proj for projection onto the second factor. The subgroup H ⊂ (R ⊗K R)× was defined in Definition 3.4. Theorem 7.2. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Let S and S ∨ be the Brauer-Severi varieties given by the
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
21
minimal right and left ideals in P(Aρ ). Then there is a morphism fC : C → S with dual fC∨ : C → S ∨ such that (i) the following diagram commutes: C
∨ fC ×fC
/ S × S∨
gC
P(R)
∼ ϕρ
Segre
/ P(Aρ ) O proj / P(Aρ )
(ii) the Brauer-Severi diagram [C → S] and the class of ρ in H correspond to the same element of H 1 (K, E[n]). The theorem is proved by combining Theorem 6.1 with the next lemma. First we recall how the element ρ ∈ H and n-covering π : C → E are ¯ × with ρ = ∂γ and related. In Section 3 we fixed an element γ ∈ R defined a cocycle ξ ∈ H 1 (K, E[n]) via w(ξσ ) = σ(γ)/γ for all σ ∈ GK . Then we let π : C → E be the ξ-twist of the trivial n-covering. Thus ¯ with π = [n] ◦ φ there is an isomorphism φ : C → E defined over K −1 and σ(φ) ◦ φ = τξσ for all σ ∈ GK . ¯ ¯ → Lemma 7.3. There is an isomorphism of K-algebras β : Aρ ⊗K K ¯ making the following diagram commute. A1 ⊗K K (9)
C
gC
φ
E
gE
/ P(R)
∼ ϕρ
·γ
/ P(R)
∼ ϕ1
proj
/ P(Aρ ) _ _ _ _/ P(Aρ ) KK LLL KKτ1 ◦β LLτL1 ◦β KK β LLL KK K L& % proj τ1 / P(A1 ) / P(Matn ) _ _ _/ P(Matn )
Proof. The first square commutes by Proposition 3.5(v). We define ¯ → A1 ⊗K K ¯ to make the second square commute. It β : Aρ ⊗K K ¯ is an isomorphism of K-algebras by [3, Lemma 4.6]. We recall from Proposition 5.3 that τ1 is an isomorphism of K-algebras. Finally, since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τ1 ◦ β. Theorem 6.1 identifies the composite of the second row of (9) as λE , where we recall fE ×f ∨
Segre
λE : E −→E Pn−1 × (Pn−1 )∨ −→ P(Matn ). Since λE factors via the Segre embedding, its image belongs to the rank 1 locus of P(Matn ). Let λC = proj ◦ϕρ ◦ gC be the composite
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J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
¯ of the first row of (9). Recalling that τ1 ◦ β is an isomorphism of Kalgebras, it follows that λC has image belonging to the rank 1 locus of P(Aρ ). By Lemma 7.1 it therefore factors via the Segre embedding. In other words, there are morphisms fC : C → S and fC∨ : C → S ∨ making the diagram in the first part of Theorem 7.2 commute. It remains to show that [C → S] is a Brauer-Severi diagram, that fC∨ is dual to fC and that [C → S] and ρ correspond to the same element of H 1 (K, E[n]). ¯ ¯ → Matn (K) ¯ induces The isomorphism of K-algebras τ1 ◦ β : Aρ ⊗K K n−1 ∨ ∨ n−1 ∨ ¯ isomorphisms ψ : S → P and ψ : S → (P ) defined over K making the following diagram commute. ∨ fC ×fC
C
ψ×ψ ∨
φ
E
∨ fE ×fE
Segre
/ S × S∨
Segre
/ Pn−1 × (Pn−1 )∨
/ P(Aρ )
τ1 ◦β
/ P(Matn )
This shows that via (φ, ψ), the morphism fC : C → S is isomorphic ¯ to fE : E → Pn−1 . Hence [C → S] is a Brauer-Severi diagram. over K Since σ(φ) ◦ φ−1 is translation by ξσ this diagram is the ξ-twist of [E → Pn−1 ], and therefore the Brauer-Severi diagram corresponding to ρ. At the same time, we see that fC∨ is dual to fC (since fE∨ is dual to fE ). This completes the proof of Theorem 7.2. 8. Finding Equations for C in Pn−1 We continue to represent elements of H 1 (K, E[n]) by elements ρ ∈ H, where the subgroup H ⊂ (R ⊗K R)× was defined in Section 3. The obstruction algebra Aρ was introduced in Section 4. If ρ represents an element of H 1 (K, E[n]) with trivial obstruction, then we make use of an explicit trivialisation isomorphism of K-algebras τρ : Aρ → Matn (K). Theorem 8.1. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Suppose given an isomorphism of K-algebras τρ : Aρ → Matn (K). Then there is a morphism fC0 : C → Pn−1 with dual fC0 ∨ such that (i) the following diagram commutes: C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨
Segre
gC
P(R)
∼ ϕρ
/ P(Aρ )
∼ τρ
/ P(Matn ) O proj / P(Matn )
EXPLICIT n-DESCENT ON ELLIPTIC CURVES
23
(ii) the Brauer-Severi diagram [C → Pn−1 ] and the class of ρ in H correspond to the same element of H 1 (K, E[n]). Proof. Let S and S ∨ be the Brauer-Severi varieties given by the minimal right and left ideals in P(Aρ ). The isomorphism τρ induces Kisomorphisms ψ 0 : S → Pn−1 and ψ 0 ∨ : S ∨ → (Pn−1 )∨ . We modify the maps fC and fC∨ of Theorem 7.2 to give maps fC0 and fC0 ∨ making the following diagram commute.
C
0 ×f 0 ∨ fC C
/ Pn−1 × (Pn−1 )∨ O
Segre
ψ 0 ×ψ 0 ∨
C
∨ fC ×fC
/ S × S∨
/ P(Matn ) O τρ
Segre
/ P(Aρ )
Since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τρ . The theorem now follows by combining the above diagram with that in Theorem 7.2. In Section 3 we explained how to write down equations for C as a curve 2 of degree n2 in P(R) ∼ = Pn −1 . Theorem 8.1 now tells us how to convert these to equations for C as a curve of degree n in Pn−1 . First we use τρ to get equations for C in P(Matn ). Then we project onto the trace zero matrices. Next we write x11 , x12 , . . . , xnn for our coordinate functions on P(Matn ) and substitute xij = xi yj , where the xi and yj are new indeterminates (2n in total). This corresponds to pulling the image of C in P(Matn ) back under the Segre map. To project to the first factor Pn−1 , we eliminate the yj . We are left with equations in the xi , and these define the image of fC0 : C → Pn−1 . We will describe in [4] how this computation can be reduced to linear algebra for any specific value of n. For aesthetic as well as practical reasons, it is desirable to find a change of coordinates on Pn−1 so that the equations for C have small coefficients. The necessary minimisation and reduction procedures will be described in a forthcoming paper [5] for the cases n = 3 and n = 4. It is also desirable to have explicit equations for the covering map π : C → E. In principle these could be obtained using the methods of Section 3. However in the cases n = 2, 3, 4 equations for π are already given by classical formulae, reproduced in [1].
24
J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL
References [1] S.Y. An, S.Y. Kim, D.C. Marshall, S.H. Marshall, W.G. McCallum and A.R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), no. 2, 304–315. [2] J.W.S. Cassels, Lectures on elliptic curves, LMS Student Texts 24, Cambridge University Press, Cambridge, 1991. [3] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, I Algebra, submitted for publication. [4] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, III Algorithms, in preparation. [5] J.E. Cremona, T.A. Fisher and M. Stoll, Minimisation and reduction for 3and 4-coverings of elliptic curves, in preparation. [6] M. Hindry and J.H. Silverman, Diophantine geometry, Graduate Texts in Mathematics 201, Springer-Verlag, New York, 2000. [7] MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24, 235–265 (1997). The Magma home page is at http://magma.maths.usyd.edu.au/magma/ [8] D. Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), pp. 29–100, Edizioni Cremonese, Rome, 1970. [9] C. O’Neil, The period-index obstruction for elliptic curves, J. Number Theory 95 (2002), no. 2, 329–339. [10] J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, New York, 1979. [11] J.H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK E-mail address: [email protected] University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK E-mail address: [email protected] Barnard College, Columbia University, Department of Mathematics, 2990 Broadway MC 4418, New York, NY 10027-6902,USA E-mail address: [email protected] ´ de Caen, Campus II - Boulevard Mare ´chal Juin, BP 5186– Universite 14032, Caen, France E-mail address: [email protected] School of Engineering and Science, International University Bremen (Jacobs University Bremen as of spring 2007), P.O. Box 750561, 28725 Bremen, Germany E-mail address: [email protected]
